In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/∞. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space $\mathcal{S}^{\prime}$ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a $\mathcal{S}^{\prime}$ -valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes.
Publié le : 2008-12-15
Classification:
Measure-valued Markov process,
fluid limit,
central limit theorem,
pure delay system,
queueing theory,
60F17,
60K25,
60B12
@article{1227708915,
author = {Decreusefond, Laurent and Moyal, Pascal},
title = {A functional central limit theorem for the M/GI/$\infty$ queue},
journal = {Ann. Appl. Probab.},
volume = {18},
number = {1},
year = {2008},
pages = { 2156-2178},
language = {en},
url = {http://dml.mathdoc.fr/item/1227708915}
}
Decreusefond, Laurent; Moyal, Pascal. A functional central limit theorem for the M/GI/∞ queue. Ann. Appl. Probab., Tome 18 (2008) no. 1, pp. 2156-2178. http://gdmltest.u-ga.fr/item/1227708915/